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Electric Current and Circuit Electric Charge (Q): Fundamental property of matter that experiences a force when in an electromagnetic field. Unit: Coulomb (C). Quantization of charge: $Q = ne$, where $n$ is an integer and $e$ is the elementary charge ($1.602 \times 10^{-19}$ C). Types: Positive (protons) and Negative (electrons). Conservation of charge: In an isolated system, the total charge remains constant. Electric Current (I): The rate of flow of electric charge through a conductor's cross-section. $I = \frac{\Delta Q}{\Delta t}$ For a steady current, $I = \frac{Q}{t}$. SI Unit: Ampere (A). 1 A = 1 C/s. Measured by an Ammeter , always connected in series to measure current through a component. Conventional Current: Direction of flow of positive charge (opposite to electron flow). Drift Velocity ($v_d$): Average velocity with which free electrons drift in a conductor under the influence of an electric field. This velocity is typically very small. Relationship between current and drift velocity: $I = n A e v_d$, where $n$ is number of charge carriers per unit volume, $A$ is cross-sectional area, $e$ is elementary charge. Current Density (J): Current per unit cross-sectional area. $J = I/A = n e v_d$. It is a vector quantity. Electric Circuit: A closed, continuous path for electric current. Essential components: Source: Battery or cell (provides potential difference). Conductor: Wires (allow charge flow with minimal resistance). Load: Resistor, bulb (consumes energy). Switch: To open or close the circuit. Circuit Symbols Component Symbol Function Electric Cell + - Source of constant potential difference Battery + - Multiple cells in series Switch (Open) Breaks the circuit, no current flow Switch (Closed) Completes the circuit, current flows Wire Joint Wires connected Wires Crossing (no join) Wires insulated from each other Electric Bulb Converts electrical energy to light and heat Resistor Opposes current flow Variable Resistor (Rheostat) Adjustable resistance Ammeter A Measures current (series connection) Voltmeter V Measures potential difference (parallel connection) Galvanometer G Detects small currents Diode Allows current in one direction only Inductor Stores energy in a magnetic field Capacitor Stores energy in an electric field Electric Potential and Potential Difference Electric Potential (V): The work done per unit positive charge by an external agent in moving the charge from a reference point (usually infinity) to a specific point in an electric field without acceleration. $V = \frac{W_{\text{infinity} \to P}}{Q}$ Potential Difference ($\Delta V$ or pd): The work done per unit positive charge to move a charge from one point to another in an electric field without acceleration. $\Delta V = V_B - V_A = \frac{W_{AB}}{Q}$ Where $W$ is work done (in Joules, J) and $Q$ is charge (in Coulombs, C). SI Unit: Volt (V). 1 Volt = 1 Joule/Coulomb. Measured by a Voltmeter , always connected in parallel across the points where potential difference is to be measured. An ideal voltmeter has infinite resistance. Electromotive Force (EMF, $\mathcal{E}$): The potential difference provided by a source (like a battery) when no current is flowing through it (open circuit). It's the maximum potential difference the source can provide, representing the energy provided per unit charge. Ohm's Law States that the current ($I$) flowing through a conductor between two points is directly proportional to the potential difference ($V$) across the two points, provided the temperature and other physical conditions remain constant. $V \propto I \implies V = IR$ Where $R$ is the constant of proportionality called Resistance . Ohmic Conductors: Materials that obey Ohm's Law (e.g., most metals at constant temperature). Their V-I graph is a straight line passing through the origin. Non-Ohmic Conductors: Materials that do not obey Ohm's Law (e.g., semiconductors, diodes, transistors). Their V-I graph is non-linear, and their resistance ($R=V/I$) is not constant. Resistance (R): The opposition offered by a material to the flow of electric current. It quantifies how much a component resists the flow of charge. SI Unit: Ohm ($\Omega$). $R = \frac{V}{I}$. Example: If a 12V battery is connected across a resistor and 2A current flows, then $R = 12V / 2A = 6\Omega$. Factors Affecting Resistance The resistance of a conductor depends on: Length (L): $R \propto L$. Longer wires have more resistance. Area of Cross-section (A): $R \propto \frac{1}{A}$. Thicker wires have less resistance. Nature of Material: Intrinsic property; some materials resist more than others. Temperature: For metals, resistance generally increases with temperature ($R_T = R_0(1 + \alpha T)$ where $\alpha$ is the temperature coefficient of resistance). For semiconductors, resistance decreases with temperature. Combining these factors, the resistance is given by: $R = \rho \frac{L}{A}$ Where $\rho$ (rho) is the resistivity of the material. Resistivity ($\rho$): An intrinsic property of a material, independent of its dimensions (length, area). It's the resistance of a wire of unit length and unit cross-sectional area. SI Unit: Ohm-meter ($\Omega \cdot m$). Conductors: Low resistivity (e.g., Copper: $1.68 \times 10^{-8} \Omega \cdot m$). Allow current to flow easily. Insulators: High resistivity (e.g., Glass: $10^{10} - 10^{14} \Omega \cdot m$). Block current flow. Semiconductors: Intermediate resistivity (e.g., Silicon: $10^3 \Omega \cdot m$). Their conductivity can be controlled. Conductance (G): The reciprocal of resistance. $G = \frac{1}{R}$. SI Unit: Siemens (S) or mho ($\mho$). Conductivity ($\sigma$): The reciprocal of resistivity. $\sigma = \frac{1}{\rho}$. SI Unit: Siemens per meter (S/m). Combination of Resistors Resistors in Series When two or more resistors are connected end-to-end, so the same current flows through each resistor sequentially. Current: Same through each resistor ($I_{total} = I_1 = I_2 = I_3$). Potential Difference: Sum of individual potential differences ($V_{total} = V_1 + V_2 + V_3$). Equivalent Resistance ($R_s$): The sum of individual resistances. This increases the total resistance. $R_s = R_1 + R_2 + R_3 + \dots$ Example: Three resistors $R_1 = 2\Omega$, $R_2 = 3\Omega$, $R_3 = 5\Omega$ are in series. Their equivalent resistance is $R_s = 2+3+5 = 10\Omega$. If a 20V battery is connected, the current is $I = V/R_s = 20V/10\Omega = 2A$. Resistors in Parallel When two or more resistors are connected across the same two points, providing multiple paths for the current to split and then recombine. Each resistor has the same potential difference across it. Current: Divides among the branches ($I_{total} = I_1 + I_2 + I_3$). Potential Difference: Same across each resistor ($V_{total} = V_1 = V_2 = V_3$). Equivalent Resistance ($R_p$): The reciprocal of the sum of the reciprocals of individual resistances. This decreases the total resistance (always less than the smallest individual resistance). $\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots$ For two resistors in parallel: $R_p = \frac{R_1 R_2}{R_1 + R_2}$ Example: Two resistors $R_1 = 6\Omega$ and $R_2 = 3\Omega$ are in parallel. Their equivalent resistance is $R_p = \frac{6 \times 3}{6 + 3} = \frac{18}{9} = 2\Omega$. If a 12V battery is connected, the total current is $I = V/R_p = 12V/2\Omega = 6A$. Internal Resistance and EMF Internal Resistance (r): The resistance offered by the electrolyte and electrodes of a cell/battery to the flow of current within itself. It causes a voltage drop inside the source. When a current $I$ is drawn from a cell with EMF $\mathcal{E}$ and internal resistance $r$, the terminal potential difference ($V$) across its terminals is less than the EMF. $V = \mathcal{E} - Ir$ (for discharging, current flows out of the positive terminal) $V = \mathcal{E} + Ir$ (for charging, current is forced into the positive terminal) The total current in a simple circuit with an external resistance $R$ is $I = \frac{\mathcal{E}}{R+r}$. Power delivered to external circuit: $P_{ext} = I^2R = (\frac{\mathcal{E}}{R+r})^2 R$. Maximum power transfer: Occurs when $R = r$. Cells in Series: Total EMF: $\mathcal{E}_{total} = \mathcal{E}_1 + \mathcal{E}_2 + \dots$ (if connected positive to negative) Total Internal Resistance: $r_{total} = r_1 + r_2 + \dots$ Cells in Parallel: (Assuming identical cells for simplicity, connected positive to positive, negative to negative) Total EMF: $\mathcal{E}_{total} = \mathcal{E}$ (same as one cell) Total Internal Resistance: $\frac{1}{r_{total}} = \frac{1}{r_1} + \frac{1}{r_2} + \dots$ (reduces internal resistance, useful for delivering higher current) Heating Effect of Electric Current (Joule's Law of Heating) When electric current flows through a conductor, electrical energy is converted into heat energy due to resistance. This phenomenon is known as the heating effect of electric current. Heat Produced (H) or Energy Dissipated (E): Directly proportional to the square of current ($I^2$). Directly proportional to the resistance of the conductor ($R$). Directly proportional to the time for which current flows ($t$). $H = I^2 R t$ Other forms of the formula (using $V=IR$): $H = V I t$ $H = \frac{V^2}{R} t$ SI Unit of Heat: Joule (J). Applications: Electric heater, electric iron, electric kettle, electric bulb (filament heats up and glows), electric fuse. Electric Fuse: A safety device that protects electrical circuits and appliances from excessive current. It consists of a wire made of a material with a low melting point (e.g., alloy of lead and tin) and high resistivity. It is always connected in series with the live wire. When current exceeds a safe limit, the fuse wire melts and breaks the circuit, preventing damage to appliances or fire hazards. Electric Power The rate at which electric energy is consumed or dissipated in an electric circuit. $P = \frac{\text{Work done}}{\text{time}} = \frac{W}{t}$ Since $W = VQ$ and $Q=It$, we have: $P = \frac{VIt}{t} = VI$ Using Ohm's Law ($V=IR$ or $I=V/R$), we can derive other forms: $P = (IR)I = I^2 R$ $P = V(\frac{V}{R}) = \frac{V^2}{R}$ SI Unit: Watt (W). 1 Watt = 1 Joule/second. Larger units: Kilowatt (kW), Megawatt (MW). 1 kW = 1000 W. Commercial Unit of Electrical Energy: Kilowatt-hour (kWh). 1 kWh = 1 unit of electricity. It is the energy consumed by an appliance of 1 kW power in 1 hour. $1 \text{ kWh} = 1000 \text{ W} \times 3600 \text{ s} = 3.6 \times 10^6 \text{ J}$. Example: A 100W bulb consumes $100 \text{ J}$ of energy per second. If it runs for 10 hours, energy consumed is $P \times t = 100 \text{ W} \times (10 \times 3600) \text{ s} = 3.6 \times 10^6 \text{ J} = 1 \text{ kWh}$. The cost would be 1 kWh multiplied by the electricity rate. Kirchhoff's Laws Essential for analyzing complex circuits that cannot be simplified by series/parallel combinations. These laws are fundamental to circuit analysis. Kirchhoff's Current Law (KCL) / Junction Rule The algebraic sum of currents entering any junction (node) in an electrical circuit is equal to the algebraic sum of currents leaving that junction. $\sum I_{in} = \sum I_{out}$ Or, $\sum I = 0$ (Currents entering are positive, leaving are negative). Based on the conservation of charge. Charge cannot accumulate at a junction. Example: If $I_1$ and $I_2$ enter a junction, and $I_3$ and $I_4$ leave, then $I_1 + I_2 = I_3 + I_4$. If $I_1 = 3A$, $I_2 = 2A$, $I_3 = 1A$, then $I_4 = I_1+I_2-I_3 = 3+2-1 = 4A$. Kirchhoff's Voltage Law (KVL) / Loop Rule The algebraic sum of all potential differences (voltages) around any closed loop in a circuit is zero. This means that if you start at a point in a closed loop and travel around it, returning to the same starting point, the total change in potential must be zero. $\sum V = 0$ Based on the conservation of energy. The potential energy of a charge must return to its initial value after completing a closed loop. Sign Convention: When traversing a resistor in the direction of current, the potential difference is $-IR$ (potential drops). Against the current, it's $+IR$ (potential rises). When traversing a battery from negative to positive terminal, the potential difference is $+\mathcal{E}$ (potential rises). From positive to negative, it's $-\mathcal{E}$ (potential drops). Example: In a simple series circuit with a battery $\mathcal{E}$ and two resistors $R_1, R_2$, starting from the negative terminal of the battery and going clockwise: $+\mathcal{E} - IR_1 - IR_2 = 0$. Wheatstone Bridge A circuit used for precise measurement of an unknown resistance. It is often used in sensor applications where a change in resistance needs to be detected. It consists of four resistors $R_1, R_2, R_3, R_4$ arranged in a diamond shape, a galvanometer (or sensitive voltmeter), and a battery. Balanced Condition: When no current flows through the galvanometer ($I_g = 0$), the bridge is balanced. This implies that the potential at points B and D (referring to a standard diamond bridge diagram) are equal. $\frac{R_1}{R_2} = \frac{R_3}{R_4}$ If $R_1$ is known, $R_2$ is adjustable (rheostat), $R_3$ is a standard resistor, then $R_4$ (unknown) can be found: $R_4 = R_3 \frac{R_2}{R_1}$. Used in strain gauges, temperature sensors, etc., where a small change in resistance needs to be measured accurately. Meter Bridge (Slide Wire Bridge) A practical application of the Wheatstone bridge, used to find the unknown resistance of a wire and compare resistances. It's a common laboratory experiment. It uses a 1-meter long wire of uniform cross-section (usually made of manganin or constantan to minimize temperature effects on resistance) and a sliding contact (jockey) to find a null point (balanced condition). The ratio arms are represented by the lengths of the wire segments. Let the unknown resistance be $R$ and the known resistance be $S$. A balancing length $L_1$ is found where the galvanometer shows no deflection. The remaining length of the wire is $100 - L_1$ (if the total wire length is 100 cm). $\frac{R}{S} = \frac{L_1}{100 - L_1}$ From this, the unknown resistance $R$ can be calculated: $R = S \frac{L_1}{100 - L_1}$. The wire must have uniform thickness to ensure its resistance per unit length is constant. The null point method is preferred because it avoids current flowing through the galvanometer, making it more accurate. Potentiometer A versatile instrument used for measuring potential differences, comparing EMFs of cells, and determining the internal resistance of a cell, without drawing any current from the source being measured (null deflection method). This makes it highly accurate for voltage measurements. Principle of Potentiometer: When a constant current flows through a wire of uniform cross-section and uniform material, the potential difference across any segment of the wire is directly proportional to its length. $V \propto L \implies V = kL$, where $k$ is the potential gradient (potential drop per unit length of the potentiometer wire). The potential gradient $k = \frac{V_{wire}}{L_{wire}}$, where $V_{wire}$ is the potential difference across the entire potentiometer wire and $L_{wire}$ is its total length. Applications: Comparison of EMFs of two cells ($\mathcal{E}_1, \mathcal{E}_2$): By finding the balancing lengths $L_1$ and $L_2$ for two cells, their EMFs can be compared: $\frac{\mathcal{E}_1}{\mathcal{E}_2} = \frac{L_1}{L_2}$ This method is more accurate than using a voltmeter because the potentiometer draws no current from the cell at the null point. Determination of Internal Resistance ($r$) of a cell: First, find the balancing length $L_1$ for the cell's EMF ($\mathcal{E}$) when it's in open circuit (no current drawn). So $\mathcal{E} = kL_1$. Then, connect the cell to an external resistance $R$ and find the balancing length $L_2$ for the terminal potential difference ($V$) across $R$. So $V = kL_2$. Since $V = \mathcal{E} - Ir$ and $I = V/R$, we can derive the internal resistance: $r = R \left( \frac{L_1 - L_2}{L_2} \right)$ Measurement of a small potential difference: A potentiometer can measure small potential differences with high accuracy, unlike voltmeters which have finite resistance and draw some current. Capacitance Capacitor: A device that stores electrical energy in an electric field. It consists of two conducting plates separated by an insulating material called a dielectric. Capacitance (C): The ability of a capacitor to store electric charge. It is defined as the ratio of the charge stored on either plate to the potential difference between the plates. $C = \frac{Q}{V}$ SI Unit: Farad (F). 1 Farad = 1 Coulomb/Volt. Farad is a very large unit, so microfarads ($\mu F = 10^{-6} F$) and picofarads ($pF = 10^{-12} F$) are commonly used. Parallel Plate Capacitor For a parallel plate capacitor with plate area $A$ and separation $d$, filled with a dielectric of permittivity $\varepsilon$: $C = \frac{\varepsilon A}{d} = \frac{\kappa \varepsilon_0 A}{d}$ $\varepsilon_0$ is the permittivity of free space ($8.85 \times 10^{-12} F/m$). $\kappa$ (kappa) is the dielectric constant (relative permittivity) of the material between the plates. For vacuum, $\kappa = 1$. Energy Stored in a Capacitor The electrical potential energy stored in a capacitor is given by: $U = \frac{1}{2} Q V = \frac{1}{2} C V^2 = \frac{1}{2} \frac{Q^2}{C}$ This energy is stored in the electric field between the plates. Combination of Capacitors Capacitors in Series: When capacitors are connected in series, the charge on each capacitor is the same, and the total potential difference is the sum of individual potential differences. $\frac{1}{C_s} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \dots$ The equivalent capacitance is always less than the smallest individual capacitance. Useful for increasing the voltage rating or reducing capacitance. Capacitors in Parallel: When capacitors are connected in parallel, the potential difference across each capacitor is the same, and the total charge is the sum of individual charges. $C_p = C_1 + C_2 + C_3 + \dots$ The equivalent capacitance is always greater than the largest individual capacitance. Useful for increasing the total charge storage capacity. Electromagnetism Magnetic Field (B) A region around a magnetic material or a moving electric charge within which the force of magnetism acts. It is a vector quantity. SI Unit: Tesla (T). 1 T = $1 \text{ N/(A} \cdot \text{m})$. Another unit is Gauss (G), 1 T = $10^4$ G. Magnetic field lines originate from the North pole and end at the South pole outside the magnet, forming continuous closed loops. Magnetic Force on a Moving Charge (Lorentz Force) A charged particle moving in a magnetic field experiences a force. $\vec{F} = q (\vec{v} \times \vec{B})$ Magnitude: $F = q v B \sin\theta$ Where $q$ is the charge, $v$ is its velocity, $B$ is the magnetic field strength, and $\theta$ is the angle between $\vec{v}$ and $\vec{B}$. Direction: Given by the right-hand rule (for positive charge). If $\theta = 0^\circ$ or $180^\circ$, $F=0$. If $\theta = 90^\circ$, $F = qvB$ (maximum force). Magnetic force does no work because it is always perpendicular to the velocity of the charge. It changes the direction of motion, not the speed. Magnetic Force on a Current-Carrying Conductor A straight conductor of length $L$ carrying current $I$ in a uniform magnetic field $B$ experiences a force: $\vec{F} = I (\vec{L} \times \vec{B})$ Magnitude: $F = I L B \sin\theta$ Where $\theta$ is the angle between the direction of current and the magnetic field. Direction: Also given by the right-hand rule. Magnetic Field Due to Current (Biot-Savart Law) This law describes the magnetic field generated by a steady current. For a small current element $I d\vec{l}$ at a distance $r$ from the element: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$ Where $\mu_0$ is the permeability of free space ($4\pi \times 10^{-7} \text{ T} \cdot \text{m/A}$). Applications: Straight current-carrying wire: $B = \frac{\mu_0 I}{2\pi r}$ (at distance $r$ from the wire). Direction by right-hand thumb rule. Circular loop at center: $B = \frac{\mu_0 I}{2R}$ (for a single turn of radius $R$). For $N$ turns, $B = \frac{\mu_0 N I}{2R}$. Solenoid: $B = \mu_0 n I$ (inside a long solenoid, where $n$ is turns per unit length). Ampere's Circuital Law The line integral of the magnetic field $\vec{B}$ around any closed loop is proportional to the total steady current $I_{enc}$ passing through the loop. $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$ Useful for calculating magnetic fields in situations with high symmetry (e.g., long straight wire, solenoid, toroid). Torque on a Current Loop in a Magnetic Field A rectangular loop of area $A$ carrying current $I$ in a uniform magnetic field $B$ experiences a torque: $\vec{\tau} = \vec{M} \times \vec{B}$ Magnitude: $\tau = M B \sin\alpha$ Where $\vec{M} = N I \vec{A}$ is the magnetic dipole moment of the loop (for $N$ turns). $\vec{A}$ is the area vector, perpendicular to the loop plane. $\alpha$ is the angle between the magnetic dipole moment vector $\vec{M}$ and the magnetic field $\vec{B}$. This principle is used in electric motors and galvanometers. Moving Coil Galvanometer A device used to detect and measure small electric currents. It works on the principle that a current-carrying coil placed in a magnetic field experiences a torque. Deflection ($\phi$) is proportional to current ($I$): $\phi \propto I$. Current sensitivity: $\frac{\phi}{I} = \frac{NAB}{k}$ (where $k$ is torsional constant). Voltage sensitivity: $\frac{\phi}{V} = \frac{NAB}{kR}$ (where $R$ is galvanometer resistance). Ammeter Conversion: A galvanometer can be converted into an ammeter by connecting a low resistance (shunt) in parallel with it. Shunt resistance $R_S = \frac{I_g G}{I - I_g}$, where $I_g$ is full-scale deflection current, $G$ is galvanometer resistance, $I$ is total current to be measured. Voltmeter Conversion: A galvanometer can be converted into a voltmeter by connecting a high resistance in series with it. Series resistance $R_{series} = \frac{V}{I_g} - G$, where $V$ is total voltage to be measured. Electromagnetic Induction (EMI) The phenomenon where an electromotive force (EMF) is induced across an electrical conductor in a changing magnetic field. Magnetic Flux ($\Phi_B$) A measure of the total number of magnetic field lines passing through a given area. $\Phi_B = \int \vec{B} \cdot d\vec{A} = B A \cos\theta$ Where $\theta$ is the angle between the magnetic field vector $\vec{B}$ and the area vector $\vec{A}$. SI Unit: Weber (Wb). 1 Wb = 1 T $\cdot m^2$. Faraday's Laws of Electromagnetic Induction First Law: Whenever magnetic flux linked with a coil changes, an EMF is induced in the coil. Second Law: The magnitude of the induced EMF is directly proportional to the rate of change of magnetic flux linked with the coil. $\mathcal{E} = -N \frac{d\Phi_B}{dt}$ Where $N$ is the number of turns in the coil. The negative sign is due to Lenz's Law . Lenz's Law The direction of the induced EMF (and hence induced current) is such that it opposes the cause producing it (i.e., the change in magnetic flux). This law is a consequence of the conservation of energy. Motional EMF The EMF induced across a conductor moving in a magnetic field. $\mathcal{E} = (v B L) \sin\theta$ For a straight conductor of length $L$ moving with velocity $v$ perpendicular to a uniform magnetic field $B$, the induced EMF is $\mathcal{E} = B L v$. Self-Induction The phenomenon where a changing current in a coil induces an EMF in the same coil. This induced EMF opposes the change in current. $\mathcal{E}_L = -L \frac{dI}{dt}$ Where $L$ is the self-inductance of the coil. SI Unit of Inductance: Henry (H). 1 H = 1 V $\cdot$ s/A. Energy stored in an inductor: $U_L = \frac{1}{2} L I^2$. Mutual Induction The phenomenon where a changing current in one coil induces an EMF in a neighboring coil. $\mathcal{E}_2 = -M \frac{dI_1}{dt}$ Where $M$ is the mutual inductance between the two coils. Mutual inductance depends on the geometry of the coils, number of turns, and their relative orientation and distance. Alternating Current (AC) An electric current that periodically reverses direction, in contrast to direct current (DC) which flows only in one direction. Instantaneous current: $I = I_0 \sin(\omega t + \phi)$ Instantaneous voltage: $V = V_0 \sin(\omega t)$ $I_0, V_0$: Peak current and voltage. $\omega = 2\pi f$: Angular frequency. $f$ is frequency (Hz). $\phi$: Phase difference between current and voltage. RMS Values Root Mean Square (RMS) values are used to represent the effective or average value of AC quantities, allowing for direct comparison with DC power. $I_{rms} = \frac{I_0}{\sqrt{2}} \approx 0.707 I_0$ $V_{rms} = \frac{V_0}{\sqrt{2}} \approx 0.707 V_0$ AC Circuits with R, L, C Resistor (R): Current and voltage are in phase ($\phi = 0$). $V_R = I R$ Inductor (L): Voltage leads current by $90^\circ$ ($\phi = +\pi/2$). Inductive Reactance: $X_L = \omega L = 2\pi f L$. $V_L = I X_L$. Capacitor (C): Current leads voltage by $90^\circ$ ($\phi = -\pi/2$). Capacitive Reactance: $X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$. $V_C = I X_C$. RLC Series Circuit Impedance (Z): The total opposition to current flow in an AC circuit. $Z = \sqrt{R^2 + (X_L - X_C)^2}$ Ohm's Law for AC: $V = I Z$ Phase Angle ($\phi$): $\tan\phi = \frac{X_L - X_C}{R}$ If $X_L > X_C$, circuit is inductive, voltage leads current. If $X_C > X_L$, circuit is capacitive, current leads voltage. Resonance in RLC Series Circuit Occurs when $X_L = X_C$. At resonance, the impedance is minimum ($Z=R$), and the current is maximum. Resonant frequency: $f_0 = \frac{1}{2\pi \sqrt{LC}}$ At resonance, the circuit behaves purely resistively, and the phase angle $\phi = 0$. Power in AC Circuits Average Power: $P_{avg} = V_{rms} I_{rms} \cos\phi$ $\cos\phi$ is the power factor . It represents the fraction of apparent power that is real power. For a purely resistive circuit, $\phi=0$, $\cos\phi=1$, $P_{avg} = V_{rms} I_{rms}$. For a purely inductive or capacitive circuit, $\phi=\pm 90^\circ$, $\cos\phi=0$, $P_{avg} = 0$ (no power dissipated). Key Concepts Summary Quantity Formula SI Unit Measurement Device Connection Charge (Q) $Q = It$, $Q = ne$ Coulomb (C) - - Current (I) $I = Q/t$, $I = nAe v_d$ Ampere (A) Ammeter Series Potential Difference (V) $V = W/Q$, $V = IR$ Volt (V) Voltmeter Parallel Resistance (R) $R = V/I$, $R = \rho L/A$ Ohm ($\Omega$) Ohmmeter - Resistivity ($\rho$) $\rho = RA/L$ Ohm-meter ($\Omega \cdot m$) - - Conductance (G) $G = 1/R$ Siemens (S) - - Conductivity ($\sigma$) $\sigma = 1/\rho$ S/m - - Equivalent Resistance (Series) $R_s = R_1 + R_2 + \dots$ Ohm ($\Omega$) - - Equivalent Resistance (Parallel) $1/R_p = 1/R_1 + 1/R_2 + \dots$ Ohm ($\Omega$) - - Heat (H) $H = I^2Rt = VIt = V^2t/R$ Joule (J) - - Power (P) $P = VI = I^2R = V^2/R$ Watt (W) Wattmeter - Electrical Energy (E) $E = Pt = VIt$ (in Joules) Joule (J) or kWh Energy Meter - Internal Resistance (r) $V = \mathcal{E} - Ir$ Ohm ($\Omega$) Potentiometer - Capacitance (C) $C = Q/V$, $C = \kappa \varepsilon_0 A/d$ Farad (F) Capacitance Meter - Stored Energy (Capacitor) $U = \frac{1}{2}CV^2$ Joule (J) - - Equivalent Capacitance (Series) $1/C_s = 1/C_1 + 1/C_2 + \dots$ Farad (F) - - Equivalent Capacitance (Parallel) $C_p = C_1 + C_2 + \dots$ Farad (F) - - Magnetic Force (Charge) $F = qvB\sin\theta$ Newton (N) - - Magnetic Force (Wire) $F = ILB\sin\theta$ Newton (N) - - Magnetic Field (Wire) $B = \frac{\mu_0 I}{2\pi r}$ Tesla (T) Teslameter - Magnetic Field (Solenoid) $B = \mu_0 n I$ Tesla (T) Teslameter - Magnetic Flux ($\Phi_B$) $\Phi_B = BA\cos\theta$ Weber (Wb) Fluxmeter - Induced EMF (Faraday's) $\mathcal{E} = -N \frac{d\Phi_B}{dt}$ Volt (V) - - Motional EMF $\mathcal{E} = BLv$ Volt (V) - - Self-Inductance (L) $\mathcal{E}_L = -L \frac{dI}{dt}$ Henry (H) LCR Meter - Stored Energy (Inductor) $U_L = \frac{1}{2}LI^2$ Joule (J) - - Inductive Reactance ($X_L$) $X_L = \omega L$ Ohm ($\Omega$) - - Capacitive Reactance ($X_C$) $X_C = 1/(\omega C)$ Ohm ($\Omega$) - - Impedance (Z) $Z = \sqrt{R^2 + (X_L - X_C)^2}$ Ohm ($\Omega$) LCR Meter - Resonant Frequency ($f_0$) $f_0 = \frac{1}{2\pi \sqrt{LC}}$ Hertz (Hz) - - Average AC Power $P_{avg} = V_{rms} I_{rms} \cos\phi$ Watt (W) Wattmeter -